Chapter 15 – Multiple Integrals
15.5 Applications of Double Integrals
Objectives:
 Understand the physical
applications of double integrals
Dr. Erickson
15.5 Applications of Double Integrals
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Applications of Double Integrals

In this section, we explore physical applications—such
as computing:
◦ Mass
◦ Electric charge
◦ Center of mass
◦ Moment of inertia
Dr. Erickson
15.5 Applications of Double Integrals
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Density and Mass

In Section 8.3, we used single integrals to compute moments
and the center of mass of
a thin plate or lamina with constant density.

Now, equipped with the double integral, we can consider a
lamina with variable density.

Suppose the lamina occupies a region D
of the xy-plane.

Also, let its density (in units of mass per unit area) at a point
(x, y) in D be given by ρ(x, y), where ρ is a continuous
function on D.
Dr. Erickson
15.5 Applications of Double Integrals
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Mass

This means that:
 ( x , y )  lim

m
A
where:
◦ Δm and ΔA are
the mass and area
of a small rectangle
that contains (x, y).
◦ The limit is taken as
the dimensions of the
rectangle approach 0.
Dr. Erickson
15.5 Applications of Double Integrals
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Mass

If we now increase the number of subrectangles, we
obtain the total mass m of the lamina as the limiting
value of the approximations:
k
m  lim
k ,l  

l

i 1
 ( x ij , y ij )  A
*
*
j 1
  ( x , y ) dA
D
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15.5 Applications of Double Integrals
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Density and Mass

Physicists also consider other types of density that can
be treated in the same manner.

For example, an electric charge is distributed over a
region D and the charge density (in units of charge per
unit area) is given by σ(x, y) at a point (x, y) in D.
Dr. Erickson
15.5 Applications of Double Integrals
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Total Charge

Then, the total charge Q is given by:
Q 
  ( x , y ) d A
D
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15.5 Applications of Double Integrals
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Example 1

Electric charge is distributed over the disk
x2 + y2  4 so that the charge density at
(x, y) is (x, y) is (x, y) = x + y + x2 + y2
(measured in coulombs per square meter).
Find the total charge on the disk.
Dr. Erickson
15.5 Applications of Double Integrals
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Moments and Centers of Mass

In Section 8.3, we found the center of mass of a lamina
with constant density.

Here, we consider a lamina with variable density.

Suppose the lamina occupies a region D and has density
function ρ(x, y).
◦ Recall from Chapter 8 that we defined
the moment of a particle about an axis as
the product of its mass and its directed distance from
the axis.
Dr. Erickson
15.5 Applications of Double Integrals
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Moments and Center of Mass

We divide D into small rectangles as earlier.

Then, the mass of Rij is approximately:
ρ(xij*, yij*) ∆A

So, we can approximate the moment of Rij with respect
to the x-axis by:
[ρ(xij*, yij*) ∆A] yij*
Dr. Erickson
15.5 Applications of Double Integrals
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Moment about the x-axis

If we now add these quantities and take
the limit as the number of sub rectangles
becomes large, we obtain the moment of
the entire lamina about the x-axis:
m
M
x
 lim
m,n 

n

i 1
y  ( x , y ) A
*
ij
*
ij
*
ij
j 1
 y  ( x , y ) dA
D
Dr. Erickson
15.5 Applications of Double Integrals
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Moment about the y-axis

Similarly, the moment about the y-axis is:
m
M
y
 lim
m ,n 

n
x
i 1
*
ij
 ( x , y ) A
*
ij
*
ij
j 1
 x  ( x , y ) dA
D
Dr. Erickson
15.5 Applications of Double Integrals
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Center of Mass



As before, we define the center of mass ( x , y )
so
that m x  M y and m y  M x .
The physical significance is that the lamina behaves as if
its entire mass is concentrated at its center of mass.
Thus, the lamina balances horizontally when supported
at its center of mass.
Dr. Erickson
15.5 Applications of Double Integrals
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Center of Mass

x 
The coordinates ( x , y ) of the center of mass of a lamina
occupying the region D and having density function ρ(x,
y) are:
M
y
m

1
x  ( x , y ) dA

m
y 
M
m
D
x

1
y  ( x , y ) dA

m
D
where the mass m is given by:
m 
  ( x , y ) d A
D
Dr. Erickson
15.5 Applications of Double Integrals
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Example 2

Find the mass and center of mass of the
lamina that occupies the region D and has
the given density function .
D is the triangular region enclosed by th e lines
x  0, y  x , and 2 x  y  6;   x , y   x
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15.5 Applications of Double Integrals
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Moment of Inertia

The moment of inertia (also called the second moment)
of a particle of mass m about an axis is defined to be
mr2, where r is the distance
from the particle to the axis.
◦ We extend this concept to a lamina with density
function ρ(x, y) and occupying a region D by
proceeding as we did for ordinary moments.
Dr. Erickson
15.5 Applications of Double Integrals
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Moment of Inertia (x-axis)

The result is the moment of inertia of the lamina about
the x-axis:
m
I x  lim
m,n 

 y
2
n
  (y
i 1
*
ij
)  (x , y )  A
2
*
ij
*
ij
j 1
 ( x , y ) dA
D
Dr. Erickson
15.5 Applications of Double Integrals
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Moment of Inertia (y-axis)

Similarly, the moment of inertia about the y-axis is given
by:
m
I y  lim
m ,n 

 x
2
n

i 1
( x ij )  ( x ij , y ij )  A
*
2
*
*
j 1
 ( x , y ) dA
D
Dr. Erickson
15.5 Applications of Double Integrals
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Moment of Inertia (Origin)

It is also of interest to consider the moment of inertia
about the origin (also called the polar moment of
inertia):
m
I 0  lim
m ,n 

 ( x
2
n
  [( x
i 1
*
ij
)  ( y ) ] ( x , y )  A
2
*
ij
2
*
ij
*
ij
j 1
 y )  ( x , y ) dA
2
D
◦ Note that I0 = Ix + Iy.
Dr. Erickson
15.5 Applications of Double Integrals
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Example 3

Find the moments of inertia Ix , Iy , Io for
the lamina of the problem below.
D is bounded by e , y  0, x  0, and x  1;
x
  x, y   y
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15.5 Applications of Double Integrals
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Example 4

A lamina occupies the part of the disk
x2 + y2  1 in the first quadrant. Find its
center of mass if the density at any point
is proportional to its distance from the
x-axis.
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15.5 Applications of Double Integrals
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Radius of Gyration
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15.5 Applications of Double Integrals
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Radius of Gyration
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Example 5

A lamina with constant density (x, y) = 
occupies the given region. Find the
moments of inertia Ix , Iy , Io and the radii
of gyration.
The part of the disk x2 + y2 ≤ a2 in the first
quadrant.
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15.5 Applications of Double Integrals
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Probability and Expected Values

Please read through these pages in your
text book.
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15.5 Applications of Double Integrals
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Dr. Erickson
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More Examples
The video examples below are from section
15.5 in your textbook. Please watch them
on your own time for extra instruction.
Each video is about 2 minutes in length.
◦ Example 2
◦ Example 3
◦ Example 4
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15.5 Applications of Double Integrals
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Demonstrations

Feel free to explore these demonstrations
below.
◦ Center of Mass of a Polygon
◦ Moment of Inertia
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15.5 Applications of Double Integrals
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